Question

Use the image to answer the question.

There are four rows of numbers with horizontal lines between the numbers and vertical lines extending to the numbers in the row below, indicating relationships.

The successive differences of a sequence are shown in the image. What is the standard form of its related polynomial expression?

(1 point)
Responses

An3+Bn2+Cn+D
upper A n cubed plus upper B n squared plus upper C n plus upper D

An4+Bn3+Cn2+Dn+E
upper A n superscript 4 baseline plus upper B n cubed plus upper C n squared plus upper D n plus E

An2+Bn+C
upper A n squared plus upper B n plus upper C

An+B

Answer 1

To determine the correct polynomial expression based on the given information about the successive differences of a sequence, we need to understand how the order of the polynomial relates to the number of rows of differences.

1. If the sequence is a polynomial of degree \( n \), the number of rows of differences until reaching a constant (the last row) will be \( n + 1 \).
2. If you have four rows of differences, it indicates that the original sequence is a polynomial of degree \( n = 3 \) (since 4 rows imply the origin degree is \( n = 3 \)).

Therefore, based on the context provided, the polynomial that corresponds to a sequence with four rows of differences is:

**\( An^3 + Bn^2 + Cn + D \)**

So the response is:

**An³ + Bn² + Cn + D**